A structured solution framework for fuzzy minimum spanning tree problem and its variants under different criteria

This paper develops a unified and structured solution framework for the minimum spanning tree (MST) problem and its variants (e.g., constrained MST problem and inverse MST problem) on networks with fuzzy link weights. It is applicable to any additive decision criterion under fuzziness (e.g., expected value, value at risk, and conditional value at risk), for generalized cases that the link weights may be represented by arbitrary types of fuzzy variables. It also applies to the entropy criterion while the link weights are continuous fuzzy variables. Following the optimality conditions of the fuzzy MST under different decision criteria proved first in this paper, it is shown that the MST problem and its variants on a fuzzy network can be converted into equivalent deterministic counterparts on their corresponding crisp networks. Consequently, these problems can be effectively solved via their deterministic counterparts without fuzzy simulation, and meanwhile, the performance of the trees under a specified criterion is precisely measured. The accuracy and efficiency are both significantly improved compared with other fuzzy simulation-based approaches. Numerical examples illustrate the superiority of the proposed solution framework. Furthermore, some new theoretical conclusions on the MST problem under fuzziness are also presented.

     

An exact algorithm for the node weighted Steiner tree problem

The Node Weighted Steiner Tree Problem (NW-STP) is a generalization of the Steiner Tree Problem. A lagrangean heuristic presented in EngevallS: StrLBN: 98, and based on the work in Lucena: 92, solves the problem by relaxing an exponential family of generalized subtour elimination constraints and taking into account only the violated ones as the computation proceeds. In EngevallS: StrLBN: 98 the computational results refer to complete graphs up to one hundred vertices. In this paper, we present a branch-and-bound algorithm based on this formulation. Its performance on the instances from the literature confirms the effectiveness of the approach. The experimentation on a newly generated set of benchmark problems, more similar to the real-world applications, shows that the approach is still valid, provided that suitable refinements on the bounding procedures and a preprocessing phase are introduced. The algorithm solves to optimality all of the considered instances up to one thousand vertices, with the exception of 11 hard instances, derived from the literature of a similar problem, the Prize Collecting Steiner Tree Problem. Received: March 2005, Revised: September 2005 AMS classification: 68M10, 90C10, 90C57 This work has been partially supported by the Ministero dell'Istruzione, Universitá e Ricerca (MIUR), Italy     

构建最小k 重控制集的概率算法

点集D ⊆ V (G) 称为图G 的k 重控制集, 如果D 满足V (G) - D 中任意结点在D 中至少有k 个邻居. 在无线网络中, 最小k 重控制集(MkDS) 用以构建健壮的虚拟骨干网. 构建虚拟骨干网是无线网络中最基本也是最重要的问题. 在本文中, 我们提出一种快速的分布式概率算法来构建k重控制集. 我们构建的k 重控制集的期望大小不超过最优解的O(k²) 倍. 算法的运行时间复杂度为O((Δ logΔ+log log n)n),其中Δ = max{|D(p)|}, D(p) 是以p 为中心半径为1 的圆盘中的结点, 最大值的比较范围是给定集合中所有的p 点.     

A metaheuristic approach to the dominating tree problem

This paper considers a recently introduced NP-hard problem on graphs, called the dominating tree problem. In order to solve this problem, we develop a variable neighborhood search (VNS) based heuristic. Feasible solutions are obtained by using the set of vertex permutations that allow us to implement standard neighborhood structures and the appropriate local search procedure. Computational experiments include two classes of randomly generated test instances and benchmark test instances from the literature. Optimality of VNS solutions on small size instances is verified with CPLEX.     

An exact solution approach for the interdiction median problem with fortification

Systematic approaches to security investment decisions are crucial for improved homeland security. We present an optimization modeling approach for allocating protection resources among a system of facilities so that the disruptive effects of possible intentional attacks to the system are minimized. This paper is based upon the p-median service protocol for an operating set of p facilities. The primary objective is to identify the subset of q facilities which, when fortified, provides the best protection against the worst-case loss of r non-fortified facilities. This problem, known as the r-interdiction median problem with fortification (IMF), was first formulated as a mixed-integer program by Church and Scaparra [R.L. Church, M.P. Scaparra, Protecting critical assets: The r-interdiction median problem with fortification, Geographical Analysis 39 (2007) 129–146]. In this paper, we reformulate the IMF as a maximal covering problem with precedence constraints, which is amenable to a new solution approach. This new approach produces good approximations to the best fortification strategies. Furthermore, it provides upper and lower bounds that can be used to reduce the size of the original model. The reduced model can readily be solved to optimality by general-purpose MIP solvers. Computational results on two geographical data sets with different structural characteristics show the effectiveness of the proposed methodology for solving IMF instances of considerable size.     

An exact solution procedure for a cluster hub location problem

This paper proposes a new (MIP) model formulation and a new solution procedure for the hub network design problem under a non-restrictive policy introduced by Sung and Jin [Sung, C.S., Jin, H.W., 2001. Dual-based approach for a hub network design problem under non-restrictive policy. European Journal of Operational Research 132 (1), 88–105]. The model formulation contains significantly fewer variables so that optimal solutions for the LP-relaxation of the model can be determined for large instances using standard procedures for LP-models. Furthermore, the LP-relaxation provides very tight lower bounds. Computational results are given, which demonstrate that the new model formulation allows for solving much larger instances. It turned out that the new (exact) solution procedure, which utilises the new model formulation, is faster than the heuristic proposed by Sung and Jin (2001). It is also shown that the problem is np-hard.     

RAMP for the capacitated minimum spanning tree problem

This paper introduces dual and primal-dual RAMP algorithms for the solution of the capacitated minimum spanning tree problem (CMST). A surrogate constraint relaxation incorporating cutting planes is proposed to explore the dual solution space. In the dual RAMP approach, primal-feasible solutions are obtained by simple tabu searches that project dual solutions onto primal feasible space. A primal-dual approach is achieved by including a scatter search procedure that further exploits the adaptive memory framework. Computational results from applying the methods to a standard set of benchmark problems disclose that the dual RAMP algorithm finds high quality solutions very efficiently and that its primal-dual enhancement is still more effective.     

An exact static solution approach for the service parts end-of-life inventory problem

This paper studies the spare parts end-of-life inventory problem that happens after the discontinuation of part production. A final ordering quantity is set such that the service process is sustained until all service obligations expire. Also, the price erosion of substitutable or new generation products over time makes it economically justifiable to consider switching to an alternative service policy for repair such as swapping the old product with a new one. This requires the joint optimization of the final order quantity and the time to switch from repair to an alternative service policy. To the best of our knowledge, the problem has not been optimally solved yet either in its static or dynamic formulation. In the current paper, we solve its static version as a bi-level optimization problem. We investigate the convexity of the objective function and give a computationally efficient algorithm to find an exact optimal solution up to any given numerical error level ??>?0. We illustrate our approach on some numerical examples and compare our results with earlier works on this problem.     

Geometric algorithms for the minimum cost assignment problem

We consider the minimum-cost λ-assignment problem, which is equivalent to the minimum-weight one-to-many matching problem on a complete bipartite graph Γ = (A, B), where A and B have n and k nodes (n ? k), respectively. Formulating the problem geometrically, we given an O(kn + k^2.5n^0.5 log^1.5 n) time randomized algorithm, which is better than the existing O(kn² + n² log n) time algorithm if n > k log k.     

Vertices contained in every minimum dominating set of a tree

In this article we begin the study of the vertex subsets of a graph G which consist of the vertices contained in all, or in no, respectively, minimum dominating sets of G. We characterize these sets for trees, and also obtain results on the vertices contained in all minimum independent dominating sets of trees. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 163‐177, 1999     

An exact solution framework for a broad class of vehicle routing problems

This paper presents an exact solution framework for solving some variants of the vehicle routing problem (VRP) that can be modeled as set partitioning (SP) problems with additional constraints. The method consists in combining different dual ascent procedures to find a near optimal dual solution of the SP model. Then, a column-and-cut generation algorithm attempts to close the integrality gap left by the dual ascent procedures by adding valid inequalities to the SP formulation. The final dual solution is used to generate a reduced problem containing all optimal integer solutions that is solved by an integer programming solver. In this paper, we describe how this solution framework can be extended to solve different variants of the VRP by tailoring the different bounding procedures to deal with the constraints of the specific variant. We describe how this solution framework has been recently used to derive exact algorithms for a broad class of VRPs such as the capacitated VRP, the VRP with time windows, the pickup and delivery problem with time windows, all types of heterogeneous VRP including the multi depot VRP, and the period VRP. The computational results show that the exact algorithm derived for each of these VRP variants outperforms all other exact methods published so far and can solve several test instances that were previously unsolved.     

An exact algorithm for connected red–blue dominating set

In the Connected Red–Blue Dominating Set problem we are given a graph G whose vertex set is partitioned into two parts R and B (red and blue vertices), and we are asked to find a connected subgraph induced by a subset S of B such that each red vertex of G is adjacent to some vertex in S. The problem can be solved in O^?(²n−|B|) time by reduction to the Weighted Steiner Tree problem. Combining exhaustive enumeration when |B| is small with the Weighted Steiner Tree approach when |B| is large, solves the problem in O^?(n^1.4143). In this paper we present a first non-trivial exact algorithm whose running time is in O^?(n^1.3645). We use our algorithm to solve the Connected Dominating Set problem in O^?(n^1.8619). This improves the current best known algorithm, which used sophisticated run-time analysis via the measure and conquer technique to solve the problem in O^?(n^1.8966).     

The minimum cost shortest-path tree game

A minimum cost shortest-path tree is a tree that connects the source with every node of the network by a shortest path such that the sum of the cost (as a proxy for length) of all arcs is minimum. In this paper, we adapt the algorithm of Hansen and Zheng (Discrete Appl. Math. 65:275?C284, 1996) to the case of acyclic directed graphs to find a minimum cost shortest-path tree in order to be applied to the cost allocation problem associated with a cooperative minimum cost shortest-path tree game. In addition, we analyze a non-cooperative game based on the connection problem that arises in the above situation. We prove that the cost allocation given by an ??à la?? Bird rule provides a core solution in the former game and that the strategies that induce those payoffs in the latter game are Nash equilibrium.     

A mixed 0-1 linear programming formulation for the exact solution of the minimum linear arrangement problem

We are concerned with the exact solution of a graph optimization problem known as minimum linear arrangement (MinLA). Define the length of each edge of a graph with respect to a linear ordering of the graph vertices. Then, the MinLA problem asks for a vertex ordering that minimizes the sum of edge lengths. MinLA has several practical applications and is NP-Hard. We present a mixed 0-1 linear programming formulation of the problem, which led to fast optimal solutions for dense graphs of sizes up to n = 23.     

Heuristics for the multi-level capacitated minimum spanning tree problem

The capacitated minimum spanning tree (CMST) problem is fundamental to the design of centralized communication networks. In this paper we consider the multi-level capacitated minimum spanning tree problem, a generalization of the well-known CMST problem. Based on work previously done in the field, three heuristics are presented, addressing unit and non-unit demand cases. The proposed heuristics have been also integrated into a mixed integer programming solver. Evaluation results are presented, for an extensive set of experiments, indicating the improvements that the heuristics bring to the particular problem.     

Greedy heuristics for the diameter-constrained minimum spanning tree problem

The diameter-constrained minimum spanning tree problem is an NP-hard combinatorial optimization problem that seeks a minimum cost spanning tree with a limit D imposed upon the length of any path in the tree. We begin by presenting four constructive greedy heuristics and, secondly, we present some second-order heuristics, performing some improvements on feasible solutions, hopefully leading to better objective function values. We present a heuristic with an edge exchange mechanism, another that transforms a feasible spanning tree solution into a feasible diameter-constrained spanning tree solution, and finally another with a repetitive mechanism. Computational results show that repetitive heuristics can improve considerably over the results of the greedy constructive heuristics, but using a huge amount of computation time. To obtain computational results, we use instances of the problem corresponding to complete graphs with a number of nodes between 20 and 60 and with the value of D varying between 4 and 9.     

Parallel primal-dual methods for the minimum cost flow problem

In this paper we discuss the parallel asynchronous implementation of the classical primaldual method for solving the linear minimum cost network flow problem. Multiple augmentations and price rises are simultaneously attempted starting from several nodes with possibly outdated price and flow information. The results are then merged asynchronously subject to rather weak compatibility conditions. We show that this algorithm is valid, terminating finitely to an optimal solution. We also present computational results using an Encore MULTIMAX that illustrate the speedup that can be obtained by parallel implementation.This work supported in part by the BM/C3 Technology branch of the United States Army Strategic Defense Command.     

Aggregation approach for the minimum binary cost tension problem

The aggregation technique, dedicated to two-terminal series–parallel graphs (TTSP-graphs) and introduced lately to solve the minimum piecewise linear cost tension problem, is adapted here to solve the minimum binary cost tension problem (BCT problem). Even on TTSP-graphs, the BCT problem has been proved to be NP-complete. As far as we know, the aggregation is the only algorithm, with mixed integer programming (MIP), proposed to solve exactly the BCT problem on TTSP-graphs. A comparison of the efficiency of both methods and a heuristic is presented.